Physics D Lecture Slides Feb., 010 Sunil Sinha UCSD Physics
Thomson s Determination of e/m of the Electron In E Field alone, electron lands at D In B field alone, electron lands at E When E and B field adjusted to cancel each other s force electron lands at F e/m = 1.7588 x 10 11 C/Kg
Millikan s Measurement of Electron Charge Find charge on oil drop is always in integral multiple of some Q q e = 1.688 x 10-19 Coulombs m e = 9.1093 x 10-31 Kg Fundamental properties (finger print) of electron (similarly can measure proton properties etc)
Where are the electrons inside the atom? Early Thought: Plum pudding model Atom has a homogenous distribution of Positive charge with electrons embedded in them (atom is neutral) e - e - e - e e- - e - e - e - e - e - e - e - e - e - e - e - e - e - e - Positively charged matter or? How to test these hypotheses? Shoot bullets at the atom and watch their trajectory. What Kind of bullets? + Core Indestructible charged bullets Ionized He ++ atom = α ++ particles Q = +e, Mass M α =4amu >> m e, V α = x 10 7 m/s (non-relavistic) [charged to probe charge & mass distribution inside atom] +
Plum Pudding Model of Atom Non-relativistic mechanics (V α /c = 0.1) In Plum-pudding model, α-rays hardly scatter because Positive charge distributed over size of atom (10-10 m) M α >> M e (like moving truck hits a bicycle) predict α-rays will pass thru array of atoms with little scatter (~1 o ) Need to test this hypothesis Ernest Rutherford
Probing Within an Atom with α Particles Most α particles pass thru gold foil with no deflection SOME ( 10-4 ) scatter at LARGE angles Φ Even fewer scatter almost backwards Why
Rutherford Scattering discovered by his PhD Student (Marsden)
Rutherford Discovers Nucleus (Nobel Prize)
Force on α-particle due to heavy Nucleus Outside radius r =R, F Q/r Inside radius r < R, F q/r = Qr/R Maximum force at radius r = R! particle trajectory is hyperbolic Scattering angle is related to impact par. # kq Q $ # " $ = % & % & ' ( ' (! Impact Parameter b cot m! v!
Rutherford Scattering: Prediction and Experimental Result # n = 4 k Z e NnA $ 1 % 4!! " 4 R & m v ' Sin ( / ) ( ) # scattered Vs φ depends on : n = # of incident alpha particles N = # of nuclei/area of foil Ze = Nuclear charge K α of incident alpha beam A= detector area
Rutherford Scattering & Size of Nucleus distance of closest appoach # r size of nucleus 1 Kinetic energy of! = K = m v!! " nucleus nucleus! particle will penetrate thru a radius r until all its kinetic energy is used up to do work AGAINST the Coulomb potential of the Nucleus: 1 m v MeV = k K! =! " = 8 % r = kze K For K =7.7.MeV, Z = 13! % r = kze K Size of Nucleus = 10!! Size of Atom = 10-10 -15 Al = 4.9& 10 m m ( Ze)( e) r $ 15 m
Dimension Matters! Size of Nucleus = 10 Size of Atom = 10-10 -15 m m how are the electrons located inside an atom? How are they held in a stable fashion? (necessary condition for us to exist ) What makes up the rest of the mass of the nucleus? Why doesn t Coulomb repulsion blow the nucleus apart? All these discoveries will require new experiments and observations
Rutherford Atom & Classical Physics?
Continuous & Discrete spectra of Elements
Visible Spectrum of Sun Through a Prism
Emission & Absorption Line Spectra of Elements
Kirchhoff Experiment : D Lines in Na D lines darken noticeably when Sodium vapor introduced Between slit and prism
Emission & Absorption Line Spectrum of Elements Emission line appear dark because of photographic exposure Absorption spectrum of Na While light passed thru Na vapor is absorbed at specific λ
Spectral Observations : series of lines with a pattern Empirical observation (by trial & error) All these series can be summarized in a simple formula 1! = R # 1 n " 1 & % (,n $ f n i > n f,n i = 1,,3,4.. i ' Fitting to spectral line series data R=1.09737 ) 10 7 m "1 How does one explain this?
The Rapidly Vanishing Atom: A Classical Disaster! Not too hard to draw analogy with dynamics under another Central Force Think of the Gravitational Force between two objects and their circular orbits. Perhaps the electron rotates around the Nucleus and is bound by their electrical charge M1M F= G! k r Q Q r 1 Laws of E&M destroy this equivalent picture : Why?
Bohr s Bold Model of Atom: Semi Quantum/Classical -e +e U ( r) KE = r F e =! k r 1 m v e m e V 1. Electron in circular orbit around proton with vel=v. Only stationary orbits allowed. Electron does not radiate when in these stable (stationary) orbits 3. Orbits quantized: M e v r = n h/π (n=1,,3 ) 4. Radiation emitted when electron jumps from a stable orbit of higher energy stable orbit of lower energy E f -E i = hf =hc/λ 5. Energy change quantized f = frequency of radiation
General Two body Motion under a central force Reduced Mass of -body system -e m e +e F V reduces to r Both Nucleus & e - revolve around their common center of mass (CM) Such a system is equivalent to single particle of reduced mass µ that revolves around position of Nucleus at a distance of (e - -N) separation µ= (m e M)/(m e +M), when M>>m, µ=m (Hydrogen atom) m e Νot so when calculating Muon (m µ = 07 m e ) or equal mass charges rotating around each other (similar to what you saw in gravitation)
Allowed Energy Levels & Orbit Radii in Bohr Model 1 e E=KE+U = mev! k r Force Equality for Stable Orbit " Coulomb attraction = CP Force " KE e k r m v mev e = = k r e Total Energy E = KE+U= - k r Negative E " Bound system This much energy must be added to the system to break up the bound atom = e r Radius of Electron Orbit : mvr = n! n! " v =, mr 1 ke substitute in KE= mev = r n! " rn =, n = 1,,... # mke n = 1" Bohr Radius a 1!! 10 a0 = = 0.59$ 10 m mke In ge = = # neral rn n a0; n 1,,... Quantized orbits of rotation 0
n since Energy Level Diagram and Atomic Transitions E = K + U = r n 0 a n Interstate transition: 0 " ke r, n =quantum number " ke 13.6 En = = " ev, n = 1,,3.. # a n n + E = = hf = E " E i f n i * " ke $ 1 1 % = " a & 0 ni n ' ( f ) ke $ 1 1 % f = " ha & 0 n f n ' ( i ) 1 f ke $ 1 1 % = = "! c hca & 0 ( n f n ' i ) $ 1 1 % = R " & n f n ' ( i ) n f
Hydrogen Spectrum: as explained by Bohr E n! ke " Z = #$ % & a0 ' n Bohr s R same as the Rydberg Constant R derived empirically from photographs of the spectral series
Another Look at the Energy levels E n! ke " = #$ % & a0 ' Z n Rydberg Constant
Bohr s Atom: Emission & Absorption Spectra photon photon
Some Notes About Bohr Like Atoms Ground state of Hydrogen atom (n=1) E 0 = -13.6 ev Method for calculating energy levels etc applies to all Hydrogenlike atoms -1e around +Ze Examples : He +, Li ++ Energy levels would be different if replace electron with Muons Bohr s method can be applied in general to all systems under a central force (e.g. gravitational instead of Coulombic)